Finite CV Analysis

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Finite Control Volume Analysis Conservation of Mass Continuity EquationReynolds transport theorem establishes relation between system rates of change and control-volume surface and volume integralsWith B = mass and b = 1, this becomesMass is conservedContinuity equationsyssys0DM DdVDt Dt syscv csDBbdV b dADt t + V n gsys cv csDdV dV dADt t + V n gcv cs0 dV dAt + V n g Conservation of Mass Continuity EquationMass flowrate equals the product of density and volume flowratem Q AV & Fixed, Nondeforming Control VolumeExample 5.1 Seawater flows steadily through a simple conical-shaped nozzle at the end of a fire hose as illustrated in Figure. If the nozzle exit velocity must be at least 20 m/s, determine the minimum pumping capacity required in m3/s.3Answer: 0.0251 m /s Q Example 5.2 Air flows steadily between two sections in a long, straight portion of 4-in. inside diameter pipe as indicated in Figure. The uniformly distributed temperature and pressure at each section are given. If the average air velocity (nonuniform velocity distribution) at section (2) is 1000 ft/s, calculate the average air velocity at section (1).1Answer: 219 ft/s V Example 5.3 Moist air (a mixture of dry air and water vapor) enters a dehumidifier at the rate of 22 slugs/hr. Liquid water drains out of the dehumidifier at a rate of 0.5 slugs/hr. Determine the mass flowrate of the dry air and the water vapor leaving the dehumidifier.2 1 3Answer: 21.5 slugs/hr m m m & & & Example 5.4 Incompressible, laminar water flow develops in a straight pipe having radius R as indicated in Figure. At section (1), the velocity profile is uniform; the velocity is equal to a constant value U and is parallel to the pipe axis everywhere. At section (2), the velocity profile is axisymmetric and parabolic, with zero velocity at the pipe wall and a maximum value of umax at the centerline. How are U and umax related? How are the average velocity at section (2), , and umax related? 2Vmaxmax 2Answers: 2 ; 2uu U V Example 5.5 A bathtub is being filled with water from a faucet. The rate of flow from the faucet is steady at 9 gal/min. The tub volume is approximated by a rectangular space as indicated in Figure. Estimate the time rate of change of the depth of water in the tub, h/ t, in in./min at any instant.( ) ( )water water2 2Answer: 1.44 in./min10 ft 10 ftjQ Q ht A Moving, Nondeforming Control VolumeSome problems are most easily solved using a moving control volume.Continuity equation for a moving nondeforming control volume:where relative velocity (fluid velocity seen by an observer moving with the control volume) is:cv cs0 dV dAt + W n gcv W V V Example 5.6 An airplane moves forward at a speed of 971 km/hr. The frontal intake area of the jet engine is 0.80 m2 and the entering air density is 0.736 kg/m3. A stationary observerdetermines that relative to the earth, the jet engine exhaust gases move away from the engine with a speed of 1050 km/hr. The engine exhaust area is 0.558 m2, and the exhaustgas density is 0.515 kg/m3. Estimate the mass flowrate of fuel into the engine in kg/hr.fuelAnswer: 9100 kg/hr m & Example 5.7 Water enters a rotating lawn sprinkler through its base at the steady rate of 1000 ml/s. If the exit area of each of the two nozzles is 30 mm2, determine the average speed of the water leaving each nozzle, relative to the nozzle, if (a) the rotary sprinkler head is stationary, (b) the sprinkler head rotates at 600 rpm, and (c) the sprinkler head accelerates from 0 to 600 rpm.22Answer: =16.7 m/s2QWA Deforming Control VolumeSome problems are most easily solved by using a deforming control volume.Deforming control volume involves changing volume size and control surface movement.Continuity equation for a deforming control volume:For the deforming control volumewhere Vcs is the velocity of the control surface as seen by a fixed observer.cv cs0 dV dAt + W n gcs + V W V Example 5.8 A syringe is used to inoculate a cow. The plunger has a face area of 500 mm2. If the liquid in the syringe is to be injected steadily at a rate of 300 cm3/min, at what speed should the plunger be advanced? The leakage rate past the plunger is 0.10 times the volume flowrate out of the needle.2 leak1Answer: =660 mm/minp Q QVA+ Example 5.9 A bathtub is being filled with water from a faucet. The rate of flow from the faucet is steady at 9 gal/min. The tub volume is approximated by a rectangular space as indicated in Figure. Estimate the time rate of change of the depth of water in the tub, h/ t, in in./min at any instant. Solve example using a deforming control volume that includes only the water accumulating in the buthtub.( ) ( )water water2 2Answer: 1.44 in./min10 ft 10 ftjQ Q ht A Newtons Second Law Linear Momentum EquationTime rate of change of the linear momentum of the system equals sum of external forces acting on the systemWhen control volume is coincident with a system at an instant of time: Reynolds transport theorem with Bsys = system momentum and b = V, becomesTime rate of change of the linear momentum of the system equals the time rate of change of the linear momentum of the contents of the control volume plus net rate of flow of linear momentum trough the control surfaceLinear momentum equation isBoth surface and body forces act on the contents of the control volumesys cv csDdV dV dADt t + V V V V n gcvcv csdV dAt + V V V n F gsyssysDdVDt V Fsys cv F F Example 5.10 A horizontal jet of water exits a nozzle with a uniform speed of V1 = 10 ft/s, strikes a vane, and is turned through an angle . Determine the anchoring force needed to hold the vane stationary. Neglect gravity and viscous effects.( ) ( )21 121 1oAnswer: 1 cos 11.64 1 cos lb sin 11.64sin lb=0 0; 090 11.64 lb; 11.64 lb=AxAzAx AzAx AzF AVF AVF FF F o180 23.3 lb; 0 Ax AzF F Example 5.11 Determine the anchoring force required to hold in place a conical nozzle attached to the end of a laboratory sink faucet when the water flowrate is 0.6 liter/s. The nozzle mass is 0.1 kg. The nozzle inlet and exit diameters are 16 mm and 5 mm, respectively. The nozzle axis is vertical and the axial distance between sections (1) and (2) is 30 mm. The pressure at section (1) is 464 kPa. Example 5.11 Determine the anchoring force required to hold in place a conical nozzle attached to the end of a laboratory sink faucet when the water flowrate is 0.6 liter/s. The nozzle mass is 0.1 kg. The nozzle inlet and exit diameters are 16 mm and 5 mm, respectively. The nozzle axis is vertical and the axial distance between sections (1) and (2) is 30 mm. The pressure at section (1) is 464 kPa.( )1 2 1 1 2 2Answer:77 8 NA n wF m w w W W p A p A + + + . & Example 5.12 Water flows through a horizontal, 180 pipe bend. The flow cross-sectional area is constant at a value of 0.1 ft2 through the bend. The flow velocity everywhere in the bend is axial and 50 ft/s/ The absolute pressures at the entrance and exit of the bend are 30 psia and 24 psia, respectively. Calculate the horizontal (x and y) components of the anchoring force required to hold the bend in place.( )1 2 1 1 2 2Answer:1324 lbAyF m v v p A p A + & Example 5.13 Air flows steadily between two cross sections in a long, straight portion of 4-in. inside diameter pipe as indicated in Figure, where the uniformly distributed temperature and pressure at each cross section are given. If the average air velocity at section (2) is 1000 ft/s, we found in Example 5.2 that the average air velocity at section (1) must be 219 ft/s. Assuming uniform velocity distributions at sections (1) and (2), determine the frictional force exerted by the pipe wall on the air flow between sections (1) and (2).( )( ) ( )1 1 2 2 1 1 2 22 1 2 2 1Answer:793 lbxxu m u m R p A p AR A p p m u u + + & && Example 5.14 If the flow of Example 5.4 is vertically upward, develop an expression for the fluid pressure drop that occurs between sections (1) and (2).( )21 1 2 2 1 1 2 2211 21 1Solution:3zAzw m w w dA p A R W p Aw R Wp pA A + + +& Example 5.15 A static thrust stand as sketched in Figure is to be designed for testing a jet engine. The following conditions are known for a typical test: Intake air velocity = 200 m/s; exhaust gas velocity = 500 m/s; intake cross-sectional area = 1 m2; intake static pressure = -22.5 kPa = 78.5 kPa (abs); intake static temperature = 268 K; exhaust static pressure = 0 kPa = 101 kPa (abs). Estimate the nominal thrust for which to design.( ) ( )1 1 2 2 1 1 th 2 2 atm 1 2thSolution:=83 kNu m u m p A F p A p A AF + + & & Linear Momentum Equation for a Moving CVFor a moving nonderofming control volumeFor a constant control volume velocity, Vcv and steady flow:( ) ( )cv cv cvcv csdV dAt + + + W V W V W n F gcvcs dA W W n F g Example 5.17 A vane on wheels moves with constant velocity V0 when a stream of water having a nozzle exit velocity of V1 is turned 45 by the vane as indicated in Figure (a). Note that this is the same moving vane considered in Section 4.4.6 earlier. Determine the magnitude and direction of the force, F, exerted by the stream of water on the vane surface. The speed of the water jet leaving the nozzle is 100 ft/s, and the vane is moving to the right with